Fig. to ignore something on purpose. John could have prevented the problem, but he looked the other way. By looking the other way, he actually made the problem worse.

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She just looked straight through me

to behave as if you do not see someone when you look at them, either because you do not notice them or because you are ignoring them

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My specific question: I have found those explanations in both the Cambridge and Oxford dictionaries. Nevertheless, I have just found sentence below in the book English Grammar in Use. Thus, I am wondering if the bold parts mean the same thing and/or are interchangeable?

I saw Sue in town yesterday, but she didn't see me. She was looking the other way.

The sentence you found in English Grammar in Use seems to be strange. Are you sure that book does not say "She did not see me" instead of "She wasn't seeing me"? Secondly, why do you mention looking straight through if you ask nothing about it? Please edit your question carefully to make it more clear what you are asking.
– oerkelensJan 13 '15 at 11:22

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And a quick answer: In the example question you found, you should probably read the other way as literal, meaning 1. on your linked freedictionary page.
– oerkelensJan 13 '15 at 11:23

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The passage in English Grammar in Use (at least in the 4th Edition) is an exercise where the reader/learner is to fill in two verbs. Correctly answered, the sentence reads: I saw Sue in town, but she didn't see me. She was looking the other way. Question edited accordingly, pending approval. Exercise p. 13 and answer p. 337. ISBN 978-0-521-18939-2
– Jim ReynoldsJan 13 '15 at 12:03

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Yes. You're right. I had some typo. You could see my updated version-- bold parted one.
– nimaJan 13 '15 at 12:05

Oerkelens is correct, above. In that sentence, the meaning is the literal one. #1 in the dictionary link you gave. So, no, the meanings of both phrases, whether literal or figurative, never have the same meaning.
– Jim ReynoldsJan 13 '15 at 12:13

1 Answer
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I saw Sue in town, but she didn't see me. She was looking the other way.

If the sentence is supposed to mean "ignore", then both "looking the other way" and "looking right through me" can be used somewhat interchangeably. They differ in scope: Usually "looking the other way" would be used for things/problems/behaviour, not a person. "Looking right trough" refers to people.

If the expression is to be taken literally, they are not interchangeable: - "looking the other way" would mean she had her heat (or at least her eyes) turned towards another point. - "looking straight through me" would indicate her being mentally absent or deep in thought, not noticing her surroundings.

That's not correct, Stephie. The phrases are "look the other way" and "to see right through (somebody)." They are significantly different. You correctly state that they can both have a general meaning related to to ignore, but to look the other way means to deliberately ignore, and moreover, the fact that it is deliberate is essential to the meaning. :D
– Jim ReynoldsJan 13 '15 at 12:17

Haha! I don't know! I was thinking of to see right through somebody; to see, for example, a trick, ploy, hidden agenda. So, I'm going to retract the above disagreement and rest with a cool towel on my forehead.
– Jim ReynoldsJan 13 '15 at 12:27